An important property of the magnetic field is that even though its profile changes with the coupling \(\xi \) , the magnetic flux \(\Phi \)  obtained by integrating the magnetic field over the entire two-dimensional area stays constant (i.e. it is independent of the value of \(\xi \) ). We show here that the magnetic flux depends only on the winding number \(n\)  and hence is a topological invariant. The quantity \(-\frac{A \,(a^{\prime })^2}{2 \,e^2 \,r^2}\)  appearing in the Lagrangian density (REF ) stems from the term \(-\frac{1}{4} F_{\mu \nu }F^{\mu \nu }\)  and hence is identified with \(-B_m^2/2\)  where \(B_m\)  is the magnetic field (no electric field is present hence the absence of an \(\tfrac{E^2}{2}\)  term). It follows that the magnetic field is given by \(B_m= \frac{\sqrt{A}\, a^{\prime }}{e \,r}\)  which reduces to the well-known result \(a^{\prime }/(e\,r)\)  for the magnetic field in fixed Minkowski spacetime [1]} where \(A(r)=1\)  identically.
